Area function antiderivatives

The Mathematica notebook which contains the live version of this activity is here.
Names:
Type in the names of each person in your group and save this file to your own account before proceeding farther.

Fill in the missing pieces of information below that state relationships between a function and it's derivative.
1) If the derivative is positive, then the function is
2) If the function is decreasing, then the derivative is
3) If the derivative is increasing then the function is concave
4) If the derivative is decreasing, then the function is concave

Shown below is the graph of f(x). Use the graph to answer the following questions.

1) On which intervals is f(x) positive? negative? increasing? decreasing?

positive:

negative:

increasing:

decreasing:

The commands below define the function f(x) and it's area function A(x) = , which gives the signed area under f(t) from t = 1 to t = x. Click anywhere in the command and press "shift-enter" to execute these commands. After doing this, Mathematica knows the definitions for f(x) and A(x).

Here is a plot of the area function A(x). Use the plot to answer the questions below.

On which intervals is A(x) increasing? decreasing? concave up? concave down?

increasing:

decreasing:

concave up:

concave down:

Compare your answers about A(x) increasing, decreasing, etc. to your answers about f(x) positive, negative, etc. This comparison should suggest that one of f(x) or A(x) is the derivative of the other.

Which function is the derivative?

Which is the function?

Here is a plot of both f(x) (black) and A(x) (in red). The plot again suggests that there is a function-derivative type relationship between f(x) and A(x).

For Mathematica notebook only:
Let's explore more closely the possibility that f(x) is the derivative of A(x), or equivalently that A(x) is an antiderivative of f(x). Once Mathematica knows the definitions of these functions, you can calculate values "just by asking". You click at a vertical location in the notebook, between any 2 of the rightmost cell brackets to start a new command (just start typing after you click).

Click between this text and the text above and type f[2] shift enter. You should see output which gives the value of f(2). To get A'(2), type (by starting a command) A'[2] and press shift enter. Pick 5 values of x from 0 to 5.6 and compare (one at a time) the values of f(x) and A'(x).